Integrand size = 35, antiderivative size = 153 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{4} a (2 b c+a f) x^4+\frac {1}{5} a (2 b d+a g) x^5+\frac {1}{6} a^2 h x^6+\frac {1}{7} b (b c+2 a f) x^7+\frac {1}{8} b (b d+2 a g) x^8+\frac {2}{9} a b h x^9+\frac {1}{10} b^2 f x^{10}+\frac {1}{11} b^2 g x^{11}+\frac {1}{12} b^2 h x^{12}+\frac {e \left (a+b x^3\right )^3}{9 b} \]
a^2*c*x+1/2*a^2*d*x^2+1/4*a*(a*f+2*b*c)*x^4+1/5*a*(a*g+2*b*d)*x^5+1/6*a^2* h*x^6+1/7*b*(2*a*f+b*c)*x^7+1/8*b*(2*a*g+b*d)*x^8+2/9*a*b*h*x^9+1/10*b^2*f *x^10+1/11*b^2*g*x^11+1/12*b^2*h*x^12+1/9*e*(b*x^3+a)^3/b
Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.82 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {b^2 x^7 \left (3960 c+7 x \left (495 d+440 e x+6 x^2 \left (66 f+60 g x+55 h x^2\right )\right )\right )+462 a^2 x \left (60 c+x \left (30 d+x \left (20 e+15 f x+12 g x^2+10 h x^3\right )\right )\right )+22 a b x^4 (630 c+x (504 d+5 x (84 e+x (72 f+7 x (9 g+8 h x)))))}{27720} \]
(b^2*x^7*(3960*c + 7*x*(495*d + 440*e*x + 6*x^2*(66*f + 60*g*x + 55*h*x^2) )) + 462*a^2*x*(60*c + x*(30*d + x*(20*e + 15*f*x + 12*g*x^2 + 10*h*x^3))) + 22*a*b*x^4*(630*c + x*(504*d + 5*x*(84*e + x*(72*f + 7*x*(9*g + 8*h*x)) ))))/27720
Time = 0.39 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2017, 2389, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (b x^3+a\right )^2 \left (h x^5+g x^4+f x^3+d x+c\right )dx+\frac {e \left (a+b x^3\right )^3}{9 b}\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (b^2 h x^{11}+b^2 g x^{10}+b^2 f x^9+2 a b h x^8+b (b d+2 a g) x^7+b (b c+2 a f) x^6+a^2 h x^5+a (2 b d+a g) x^4+a (2 b c+a f) x^3+a^2 d x+a^2 c\right )dx+\frac {e \left (a+b x^3\right )^3}{9 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 c x+\frac {1}{2} a^2 d x^2+\frac {1}{6} a^2 h x^6+\frac {1}{7} b x^7 (2 a f+b c)+\frac {1}{4} a x^4 (a f+2 b c)+\frac {1}{8} b x^8 (2 a g+b d)+\frac {1}{5} a x^5 (a g+2 b d)+\frac {e \left (a+b x^3\right )^3}{9 b}+\frac {2}{9} a b h x^9+\frac {1}{10} b^2 f x^{10}+\frac {1}{11} b^2 g x^{11}+\frac {1}{12} b^2 h x^{12}\) |
a^2*c*x + (a^2*d*x^2)/2 + (a*(2*b*c + a*f)*x^4)/4 + (a*(2*b*d + a*g)*x^5)/ 5 + (a^2*h*x^6)/6 + (b*(b*c + 2*a*f)*x^7)/7 + (b*(b*d + 2*a*g)*x^8)/8 + (2 *a*b*h*x^9)/9 + (b^2*f*x^10)/10 + (b^2*g*x^11)/11 + (b^2*h*x^12)/12 + (e*( a + b*x^3)^3)/(9*b)
3.4.87.3.1 Defintions of rubi rules used
Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coeff[Px, x, n - 1] *x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p , 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] && !MatchQ[Px, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ [{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 2.01 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {b^{2} h \,x^{12}}{12}+\frac {b^{2} g \,x^{11}}{11}+\frac {b^{2} f \,x^{10}}{10}+\frac {\left (2 a b h +b^{2} e \right ) x^{9}}{9}+\frac {\left (2 a b g +b^{2} d \right ) x^{8}}{8}+\frac {\left (2 a f b +b^{2} c \right ) x^{7}}{7}+\frac {\left (a^{2} h +2 a e b \right ) x^{6}}{6}+\frac {\left (a^{2} g +2 a b d \right ) x^{5}}{5}+\frac {\left (a^{2} f +2 a b c \right ) x^{4}}{4}+\frac {a^{2} e \,x^{3}}{3}+\frac {a^{2} d \,x^{2}}{2}+a^{2} c x\) | \(149\) |
| norman | \(\frac {b^{2} h \,x^{12}}{12}+\frac {b^{2} g \,x^{11}}{11}+\frac {b^{2} f \,x^{10}}{10}+\left (\frac {2}{9} a b h +\frac {1}{9} b^{2} e \right ) x^{9}+\left (\frac {1}{4} a b g +\frac {1}{8} b^{2} d \right ) x^{8}+\left (\frac {2}{7} a f b +\frac {1}{7} b^{2} c \right ) x^{7}+\left (\frac {1}{6} a^{2} h +\frac {1}{3} a e b \right ) x^{6}+\left (\frac {1}{5} a^{2} g +\frac {2}{5} a b d \right ) x^{5}+\left (\frac {1}{4} a^{2} f +\frac {1}{2} a b c \right ) x^{4}+\frac {a^{2} e \,x^{3}}{3}+\frac {a^{2} d \,x^{2}}{2}+a^{2} c x\) | \(149\) |
| gosper | \(\frac {1}{12} b^{2} h \,x^{12}+\frac {1}{11} b^{2} g \,x^{11}+\frac {1}{10} b^{2} f \,x^{10}+\frac {2}{9} a b h \,x^{9}+\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{4} x^{8} a b g +\frac {1}{8} b^{2} d \,x^{8}+\frac {2}{7} x^{7} a f b +\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{6} a^{2} h \,x^{6}+\frac {1}{3} a e b \,x^{6}+\frac {1}{5} x^{5} a^{2} g +\frac {2}{5} x^{5} d b a +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(155\) |
| risch | \(\frac {1}{12} b^{2} h \,x^{12}+\frac {1}{11} b^{2} g \,x^{11}+\frac {1}{10} b^{2} f \,x^{10}+\frac {2}{9} a b h \,x^{9}+\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{4} x^{8} a b g +\frac {1}{8} b^{2} d \,x^{8}+\frac {2}{7} x^{7} a f b +\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{6} a^{2} h \,x^{6}+\frac {1}{3} a e b \,x^{6}+\frac {1}{5} x^{5} a^{2} g +\frac {2}{5} x^{5} d b a +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(155\) |
| parallelrisch | \(\frac {1}{12} b^{2} h \,x^{12}+\frac {1}{11} b^{2} g \,x^{11}+\frac {1}{10} b^{2} f \,x^{10}+\frac {2}{9} a b h \,x^{9}+\frac {1}{9} b^{2} e \,x^{9}+\frac {1}{4} x^{8} a b g +\frac {1}{8} b^{2} d \,x^{8}+\frac {2}{7} x^{7} a f b +\frac {1}{7} b^{2} c \,x^{7}+\frac {1}{6} a^{2} h \,x^{6}+\frac {1}{3} a e b \,x^{6}+\frac {1}{5} x^{5} a^{2} g +\frac {2}{5} x^{5} d b a +\frac {1}{4} a^{2} f \,x^{4}+\frac {1}{2} a b c \,x^{4}+\frac {1}{3} a^{2} e \,x^{3}+\frac {1}{2} a^{2} d \,x^{2}+a^{2} c x\) | \(155\) |
1/12*b^2*h*x^12+1/11*b^2*g*x^11+1/10*b^2*f*x^10+1/9*(2*a*b*h+b^2*e)*x^9+1/ 8*(2*a*b*g+b^2*d)*x^8+1/7*(2*a*b*f+b^2*c)*x^7+1/6*(a^2*h+2*a*b*e)*x^6+1/5* (a^2*g+2*a*b*d)*x^5+1/4*(a^2*f+2*a*b*c)*x^4+1/3*a^2*e*x^3+1/2*a^2*d*x^2+a^ 2*c*x
Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{12} \, b^{2} h x^{12} + \frac {1}{11} \, b^{2} g x^{11} + \frac {1}{10} \, b^{2} f x^{10} + \frac {1}{9} \, {\left (b^{2} e + 2 \, a b h\right )} x^{9} + \frac {1}{8} \, {\left (b^{2} d + 2 \, a b g\right )} x^{8} + \frac {1}{7} \, {\left (b^{2} c + 2 \, a b f\right )} x^{7} + \frac {1}{6} \, {\left (2 \, a b e + a^{2} h\right )} x^{6} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{5} \, {\left (2 \, a b d + a^{2} g\right )} x^{5} + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{4} \, {\left (2 \, a b c + a^{2} f\right )} x^{4} + a^{2} c x \]
1/12*b^2*h*x^12 + 1/11*b^2*g*x^11 + 1/10*b^2*f*x^10 + 1/9*(b^2*e + 2*a*b*h )*x^9 + 1/8*(b^2*d + 2*a*b*g)*x^8 + 1/7*(b^2*c + 2*a*b*f)*x^7 + 1/6*(2*a*b *e + a^2*h)*x^6 + 1/3*a^2*e*x^3 + 1/5*(2*a*b*d + a^2*g)*x^5 + 1/2*a^2*d*x^ 2 + 1/4*(2*a*b*c + a^2*f)*x^4 + a^2*c*x
Time = 0.03 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.07 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=a^{2} c x + \frac {a^{2} d x^{2}}{2} + \frac {a^{2} e x^{3}}{3} + \frac {b^{2} f x^{10}}{10} + \frac {b^{2} g x^{11}}{11} + \frac {b^{2} h x^{12}}{12} + x^{9} \cdot \left (\frac {2 a b h}{9} + \frac {b^{2} e}{9}\right ) + x^{8} \left (\frac {a b g}{4} + \frac {b^{2} d}{8}\right ) + x^{7} \cdot \left (\frac {2 a b f}{7} + \frac {b^{2} c}{7}\right ) + x^{6} \left (\frac {a^{2} h}{6} + \frac {a b e}{3}\right ) + x^{5} \left (\frac {a^{2} g}{5} + \frac {2 a b d}{5}\right ) + x^{4} \left (\frac {a^{2} f}{4} + \frac {a b c}{2}\right ) \]
a**2*c*x + a**2*d*x**2/2 + a**2*e*x**3/3 + b**2*f*x**10/10 + b**2*g*x**11/ 11 + b**2*h*x**12/12 + x**9*(2*a*b*h/9 + b**2*e/9) + x**8*(a*b*g/4 + b**2* d/8) + x**7*(2*a*b*f/7 + b**2*c/7) + x**6*(a**2*h/6 + a*b*e/3) + x**5*(a** 2*g/5 + 2*a*b*d/5) + x**4*(a**2*f/4 + a*b*c/2)
Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{12} \, b^{2} h x^{12} + \frac {1}{11} \, b^{2} g x^{11} + \frac {1}{10} \, b^{2} f x^{10} + \frac {1}{9} \, {\left (b^{2} e + 2 \, a b h\right )} x^{9} + \frac {1}{8} \, {\left (b^{2} d + 2 \, a b g\right )} x^{8} + \frac {1}{7} \, {\left (b^{2} c + 2 \, a b f\right )} x^{7} + \frac {1}{6} \, {\left (2 \, a b e + a^{2} h\right )} x^{6} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{5} \, {\left (2 \, a b d + a^{2} g\right )} x^{5} + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{4} \, {\left (2 \, a b c + a^{2} f\right )} x^{4} + a^{2} c x \]
1/12*b^2*h*x^12 + 1/11*b^2*g*x^11 + 1/10*b^2*f*x^10 + 1/9*(b^2*e + 2*a*b*h )*x^9 + 1/8*(b^2*d + 2*a*b*g)*x^8 + 1/7*(b^2*c + 2*a*b*f)*x^7 + 1/6*(2*a*b *e + a^2*h)*x^6 + 1/3*a^2*e*x^3 + 1/5*(2*a*b*d + a^2*g)*x^5 + 1/2*a^2*d*x^ 2 + 1/4*(2*a*b*c + a^2*f)*x^4 + a^2*c*x
Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.01 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=\frac {1}{12} \, b^{2} h x^{12} + \frac {1}{11} \, b^{2} g x^{11} + \frac {1}{10} \, b^{2} f x^{10} + \frac {1}{9} \, b^{2} e x^{9} + \frac {2}{9} \, a b h x^{9} + \frac {1}{8} \, b^{2} d x^{8} + \frac {1}{4} \, a b g x^{8} + \frac {1}{7} \, b^{2} c x^{7} + \frac {2}{7} \, a b f x^{7} + \frac {1}{3} \, a b e x^{6} + \frac {1}{6} \, a^{2} h x^{6} + \frac {2}{5} \, a b d x^{5} + \frac {1}{5} \, a^{2} g x^{5} + \frac {1}{2} \, a b c x^{4} + \frac {1}{4} \, a^{2} f x^{4} + \frac {1}{3} \, a^{2} e x^{3} + \frac {1}{2} \, a^{2} d x^{2} + a^{2} c x \]
1/12*b^2*h*x^12 + 1/11*b^2*g*x^11 + 1/10*b^2*f*x^10 + 1/9*b^2*e*x^9 + 2/9* a*b*h*x^9 + 1/8*b^2*d*x^8 + 1/4*a*b*g*x^8 + 1/7*b^2*c*x^7 + 2/7*a*b*f*x^7 + 1/3*a*b*e*x^6 + 1/6*a^2*h*x^6 + 2/5*a*b*d*x^5 + 1/5*a^2*g*x^5 + 1/2*a*b* c*x^4 + 1/4*a^2*f*x^4 + 1/3*a^2*e*x^3 + 1/2*a^2*d*x^2 + a^2*c*x
Time = 0.11 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx=x^4\,\left (\frac {f\,a^2}{4}+\frac {b\,c\,a}{2}\right )+x^7\,\left (\frac {c\,b^2}{7}+\frac {2\,a\,f\,b}{7}\right )+x^5\,\left (\frac {g\,a^2}{5}+\frac {2\,b\,d\,a}{5}\right )+x^8\,\left (\frac {d\,b^2}{8}+\frac {a\,g\,b}{4}\right )+x^6\,\left (\frac {h\,a^2}{6}+\frac {b\,e\,a}{3}\right )+x^9\,\left (\frac {e\,b^2}{9}+\frac {2\,a\,h\,b}{9}\right )+\frac {a^2\,d\,x^2}{2}+\frac {a^2\,e\,x^3}{3}+\frac {b^2\,f\,x^{10}}{10}+\frac {b^2\,g\,x^{11}}{11}+\frac {b^2\,h\,x^{12}}{12}+a^2\,c\,x \]